We are given that the volume of the sphere $B$ is 5 times the volume of the cylinder $A$. We need to find an expression for the radius $r$ of the sphere in terms of the radius $R$ and height $H$ of the cylinder.

Step 1: Write the volume of the cylinder $A$

The formula for the volume of a cylinder is: $V_{\text{cylinder}} = \pi R^2 H$ where $R$ is the radius of the base of the cylinder, and $H$ is the height of the cylinder.

Step 2: Write the volume of the sphere $B$

The formula for the volume of a sphere is: $V_{\text{sphere}} = \frac{4}{3} \pi r^3$ where $r$ is the radius of the sphere.

Step 3: Relate the volumes

We are told that the volume of the sphere is 5 times the volume of the cylinder. Therefore, we have the equation: $V_{\text{sphere}} = 5 \times V_{\text{cylinder}}$ Substituting the formulas for the volumes: $\frac{4}{3} \pi r^3 = 5 \times \pi R^2 H$

Step 4: Simplify the equation

We can divide both sides of the equation by $\pi$ to eliminate it: $\frac{4}{3} r^3 = 5 R^2 H$

Step 5: Solve for $r^3$

Multiply both sides by $\frac{3}{4}$ to isolate $r^3$: $r^3 = \frac{15}{4} R^2 H$

Step 6: Solve for $r$

Take the cube root of both sides to solve for $r$: $r = \left( \frac{15}{4} R^2 H \right)^{\frac{1}{3}}$

Thus, the expression for the radius $r$ of the sphere in terms of the radius $R$ and height $H$ of the cylinder is: $r = \left( \frac{15}{4} R^2 H \right)^{\frac{1}{3}}$

Do you want more details or have any questions?

5 Related Questions:

1. How would the expression change if the sphere’s volume was 3 times that of the cylinder?
2. What is the volume of the sphere if $R = 2$ and $H = 3$?
3. How does the volume of the cylinder change if both $R$ and $H$ are doubled?
4. Can you find the surface area of the cylinder in terms of $R$ and $H$?
5. How does the relationship change if the height of the cylinder is a function of its radius, say $H = 2R$?

Tip:

When dealing with volumes of 3D shapes, always make sure to pay attention to units. Using consistent units across all measurements will help avoid mistakes in calculations.